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प्रश्न
If `tan theta = 1/sqrt(5), "write the value of" (( cosec^2 theta - sec^2 theta))/(( cosec^2 theta - sec^2 theta))`
उत्तर
` (( cosec^2 theta - sec^2 theta))/((cosec^2 theta + sec^2 theta))`
=` ((1+cot^2 theta) -( 1+ tan^2 theta))/((1+ cot^2 theta)+( 1+ tan^2 theta))`
=`((1+ 1/ tan^2 theta)-(1+ tan^2 theta))/((1+ 1/ tan^2 theta)-(1+ tan^2 theta))`
=`((1+ 1/ tan^2 theta-1- tan^2 theta))/((1+ 1/ tan^2 theta +1+ tan^2 theta))`
=` ((1/ tan^2 theta - tan^2 theta ))/((1/ tan^2 theta + tan^2 theta +2))`
=`((sqrt(5)/1)^2 - ( 1/sqrt(5))^2 )/((sqrt(5)/1)^2 + (1/sqrt(5))^2+2)`
=`((5/1+1/5))/((5/1+1/5+2/1))`
=`((24/5))/((36/5))`
=`24/36`
=`2/3`
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= `1/(sinθ xx cosθ)` ............... `square`
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∴ cotθ + tanθ = cosecθ × secθ
